This lemma is essentially equivalent to the (A) \(\Leftrightarrow \)  (B) part of [1]}. For the sake of readability, we give the proof below. We also remark that given functions \(u(X; t)\)  for \(X \in \mathbb {W}^n\)  such that they satisfy (REF ) with \(P(X; t) = u(X; t)\) , then we can construct \(u^0(X; t)\)  inductively for all \(X \in \mathbb {Z}^n\) , such that \(u^0(X; t)\)  satisfy (REF ) for all \(X \in \mathbb {Z}^n\) , and are related to \(u(X; t)\)  by (REF ) if \(X \in \mathbb {W}^n\) . The \(n = 2\)  case is discussed in detail in Remark REF , and we do not give the proof for the general \(n\)  case since it is not related to our main result.
